Probability-Probability (P-P) / Quantile-Quantile (Q-Q)

Probability-Probability (P-P) / Quantile-Quantile (Q-Q) is commonly used to visually assess whether data follow a specific distribution. In SPSSAU, the default is to check for normality. SPSSAU supports:

Batch generation of P-P or Q-Q for selected 'titles'.

P-P plot is generated by default, with Q-Q as an option.

Currently, SPSSAU only supports the normal distribution.

Q-Q Calculation Steps

P-P compares the cumulative probability of the sample data with the cumulative probability of the theoretical normal distribution. Specifically:

Sample Cumulative Probability

For sample data X₁, X₂, ..., X_n, sort the data in ascending order as X₍₁₎, X₍₂₎, ..., X_(n) and calculate the cumulative probability for each sample point:

$$P(X\le X_{(i)})=\frac{i}{n}(i=1,2,\dots ,n)$$

Theoretical Normal Distribution Cumulative Probability

Assuming the sample data follow a normal distribution, calculate the corresponding theoretical cumulative probability:

$$F(X_{(i)})=P(X\le X_{(i)})$$

$$F(X_{(i)});\mu ,\sigma )=\varphi (\frac{X_{(i)}-\mu }{\sigma })$$

Φ is the cumulative distribution function of the standard normal distribution.

Plotting P-P

In P-P, the X-axis represents P(X ≤ X_(i)), and the Y-axis represents F(X_(i)). If the sample data follow a normal distribution, the points on the P-P should lie close to the 45-degree line.

Q-Q Calculation Steps

Q-Q compares the empirical quantiles of the sample with the theoretical quantiles of the normal distribution. The calculation steps are as follows:

In Q-Q, the formula for sample quantiles includes p, which represents the quantile proportion. Specifically, p is a value between 0 and 1, used to determine the position of the desired quantile.

Sample Quantiles

For sample data X₁, X₂, ..., X_n, the sample quantile can be expressed as:

$$Q_p=X_{(i)}(i=\lceil pn\rceil )(p\in [0,1])$$

p is the proportion (a value between 0 and 1) used to determine the quantile position, and |pn| represents the ceiling function.

Theoretical Quantiles of the Normal Distribution

For a standard normal distribution (with mean 0 and standard deviation 1), the theoretical quantile is given by: Q_F^-1(p). For a general normal distribution with mean μ and standard deviation σ, the quantile is calculated as:

$$Q_{F^{-1}(p)}=\mu +\sigma \times {\varphi }^{-1}(p)$$

• μ represents the mean of the data.
• σ represents the standard deviation of the data.
• Φ^-1(p) is the inverse cumulative distribution function of the standard normal distribution.

Plotting Q-Q

In Q-Q, the X-axis typically displays the sample quantiles (Q_p), while the Y-axis displays the theoretical quantiles of the normal distribution (Q_F^-1(p)). If the sample data are normally distributed, the points on the Q-Q should lie close to the 45-degree line.